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Formal Poisson (co)homology of the Lefschetz singularity

Lauran Toussaint, Florian Zeiser

Abstract

We compute the formal Poisson cohomology groups of a real Poisson structure $π$ on $\mathbb{C}^2$ associated to the Lefschetz singularity $(z_1, z_2)\mapsto z_1^2+z_2^2$. In particular we correct an erroneous computation in the literature. The definition of $π$ depends on a choice of volume form. Using the main result we formally classify all Poisson structure arising from different choices of volume forms.

Formal Poisson (co)homology of the Lefschetz singularity

Abstract

We compute the formal Poisson cohomology groups of a real Poisson structure on associated to the Lefschetz singularity . In particular we correct an erroneous computation in the literature. The definition of depends on a choice of volume form. Using the main result we formally classify all Poisson structure arising from different choices of volume forms.
Paper Structure (27 sections, 33 theorems, 298 equations)

This paper contains 27 sections, 33 theorems, 298 equations.

Table of Contents

  1. Introduction
  2. An interpretation of the results
  3. The foliation and the Casimir functions
  4. The Lie algebra $H^1({\mathbb{R}}^4,\pi)$
  5. $H^2({\mathbb{R}}^4,\pi)$: Infinitesimal deformations
  6. Higher degrees and the algebraic structure
  7. Preliminaries
  8. Jacobi Poisson structures
  9. (Homological) Algebra
  10. Hilbert-Poincare series
  11. Regular sequences
  12. Standard bases
  13. Division groups
  14. Consequence of Proposition \ref{['proposition: iso division group']}
  15. Proposition \ref{['proposition: iso division group']}: The proof and a Corollary
  16. ...and 12 more sections

Key Result

Theorem 1.2

The formal Poisson homology groups $H_{\bullet}({\mathbb{R}}^4,\pi)$ of $({\mathbb{R}}^4,\pi)$ are uniquely described in the various degrees as follows:

Theorems & Definitions (64)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Remark 2.1
  • Corollary 2.2
  • Corollary 2.3
  • Corollary 2.4
  • Proposition 2.5
  • ...and 54 more